Problem: A weight is attached to the end of a spring. Its height after $t$ seconds is given by the equation $ h(t) = 5 - 2\sin\left(\dfrac{2\pi (t+1)}{7}\right)$. When does the weight first reach its maximum height? Give an exact answer. When $t =~$
Solution: First, let's look at when $\sin u$ achieves its minimum height. Since the coefficient of $\sin$ is negative, these will correspond to places where $h(t)$ achieves its maximum height. $\sin u$ achieves its minimum height when $u = -\dfrac{\pi}{2}$ plus a multiple of $2\pi$. So $h(t)$ reaches its maximum height when $\begin{aligned} \frac{2\pi(t+1)}{7} &= -\frac{\pi}{2} + 2\pi n\\ t+1 &= -\frac{7}{4} + 7n \\ t &= -\frac{11}{4} + 7 n \end{aligned}$ for $n$, an integer. The first positive value where $h(t)$ is at its maximum is when $n = 1$, and $t = -\dfrac{11}{4} + 7 = 4.25$. The weight first reaches its maximum height when $t = 4.25$ seconds.